St\"uckelberg interference in a superconducting qubit under periodic latching modulation

When the level separation of a qubit is modulated periodically across an avoided crossing, tunneling to the excited state - and consequently Landau-Zener-St\"uckelberg interference - can occur. The types of modulation studied so far correspond to a continuous change of the level separation. Here we study periodic latching modulation, in which the level separation is switched abruptly between two values and is kept constant otherwise. In this case, the conventional approach based on the asymptotic Landau-Zener (LZ) formula for transition probabilities is not applicable. We develop a novel adiabatic-impulse model for the evolution of the system and derive the resonance conditions. Additionally, we derive analytical results based on the rotating-wave approximation (RWA). The adiabatic-impulse model and the RWA results are compared with those of a full numerical simulation. These theoretical predictions are tested in an experimental setup consisting of a transmon whose flux bias is modulated with a square wave form. A rich spectrum is observed, with distinctive features correspoding to two regimes: slow-modulation and fast-modulation. These experimental results are shown to be in very good agreement with the theoretical models. Also, differences with respect to the well known case of sinusoidal modulation are discussed, both theoretically and experimentally.


I. INTRODUCTION
A paradigmatic example of quantum mechanical timeevolution is the Landau-Zener (LZ) problem 1 : in its modern formulation, a qubit is swept across an avoided crossing of the adiabatic energy states. The model is characterized by the asymptotic LZ probability p LZ of making a transition between the states, which is typically calculated for energy sweeps linear in time. In a coherent system, if these traversals across the crossing are repeated periodically, one observes the so-called Landau-Zener-Stückelberg (LZS) oscillations of the qubit population, caused by interference of the different evolutionary paths 2 .
The LZS interference has been realized in a variety of systems, such as Rydberg atoms 3 , superconducting qubits 4 , semiconductor quantum dots 5,6 , donors in silicon nanowires 7 , nitrogen vacancy (NV) centers in diamond 8 , nanomechanical oscillators 9 , and ultracold atoms in accelerated optical lattices 10 . In these experimental realizations, the periodic modulation between two extrema of the transition energy has been achieved by driving the qubit longitudinally with a triangular or sinusoidal signal. By assuming that the extrema of the transition energy are sufficiently far away from the crossing, one can estimate the transition probability amplitude with the asymptotic LZ probability. Accordingly, the LZ model reproduces the interference between the different evolutionary paths.
However, this theoretical description becomes inadequate if the qubit transition frequency is switched abruptly between two constant values in a periodic manner, see Fig. 1 (a). Indeed, the result of the LZS model for sudden switching would be p LZ = 1, predicting the disappearance of the characteristic interference pattern for repeated traversals. Thus a new theoretical model is needed. In the following, we will use the term 'periodic latching' to designate this type of modulation, because in-between the switches the qubit is latched onto a fixed value of the energy separation. In this case we can separate two relevant time scales, fast and slow, for switching and latching, respectively. This brings in a qualitatively new feature compared to the sinusoidal or triangular modulation where only one timescale exists (the period) for both the transition and the adiabatic evolution. As we will demonstrate, periodic latching results in Stückelberg interference patterns with specific features, qualitatively different from those obtained by standard LZ modulation.
The problem of discontinuous periodic modulation appeared for the first time in nuclear magnetic resonance experiments where the nuclear spin evolution was manipulated by periodic trains of sharp, intense pulses 11 . Recently, the problem of qubit modulation with multiple timescales has attracted renewed attention. For example, Ref. 12 considered modulations with two different Fourier components (at frequencies Ω and 2Ω, or at Ω and 3Ω). Also, aperiodic sequences of sharp pulses have been employed in superconducting circuits to create quantum simulations 13 of weak localization of electrons in disordered conductors 14 and motional averaging 15 , while bi-harmonic modulation has been employed to simulate universal quantum fluctuations 16 . Also, one notices that the effect of sudden changes in the transition frequency is similar to that produced by defects and two-level fluctuators [17][18][19] . But, to our best knowledge, the periodic latching modulation has not been previously discussed in the literature.
The periodic latching modulation can be implemented arXiv:1412.3982v1 [quant-ph] 12 Dec 2014 in a circuit-QED setup 20 , allowing us to test the theoretical predictions against the experiment. The magnetic flux threading the SQUID of the transmon 20 can be modulated with a square pulse pattern, which naturally brings in two time scales: the duty cycle provides the periodicity, while the raise and fall times occur on a different, much shorter, time-scale. However, the transition frequencies in this setup lay in the GHz range. This presents a technical challenge for the realization of the square pulses, which even with state-of-the art equipment cannot be generated and transmitted undistorted in a cryogenic setup at so high frequencies. We demonstrate here that this problem can be circumvented by driving the qubit near resonance which effectively leads, in the rotating frame, to transition frequencies in the MHz range. Rapid and precise control of the qubit's transition frequency is generally important in, e.g., the field of quantum computing 21 , and even in the study of quantum fields in curved spacetimes [22][23][24][25] .
Our results can be seen as step along this line of research, and suggest that the use of a rotating frame could be an alternative route to realizing these experiments. The paper is organized in the following way. In Sec. II, we construct an adiabatic-impulse theory appropriate for the modeling of periodic latching modulations. We also derive the excited state population in the steady state and the locations of the population extrema. Section III is devoted to our experimental realization consisting of a transmon with a flux modulation of square wave form. We show that by dressing the transmon with an additional microwave drive, the resulting effective Hamiltonian is of the generic avoided-crossing form. The discussion of the slow-modulation and fast-modulation (RWA) regimes and the comparison between the theoretical predictions and the experimental data is done in Sec. IV. Especially, we compare the experimental data with the numerical results for the transmon, including higher energy levels. Also, we compare the experimental and numerical data with those resulting from sinusoidal driving, and extract the sideband traces demonstrating the differences in the two forms of driving. In Sec. V we conclude the paper with a summary and future prospects.

II. STÜCKELBERG INTERFERENCE UNDER PERIODIC LATCHING MODULATION
Conventionally, the periodic level-crossing problem has been studied in terms of the generic Hamiltonian This represents a two-level system with off-diagonal coupling g, and transition frequency ν modulated by a timeperiodic function f (t). The types of modulation f (t) studied so far have been of sinusoidal and triangular form [3][4][5][6][7][8][9][10] . The latter case corresponds precisely to the linear time-dependence originally introduced by Landau 1 , while the former can be approximated as linear near the avoided crossing region. In both of the above cases there is only one time scale involved in f (t), that is, the period of the modulation. Accordingly, the dynamics of the time-periodic system is formally discretized into an adiabatic evolution interrupted by instantaneous nonadiabatic Landau-Zener(LZ) transitions at the avoided crossing 2 . The probability of a single transition between the adiabatic energy states is given by the celebrated Landau-Zener formula. Moreover, the periodic Landau-Zener transitions can interfere, leading to LZS-oscillations of the qubit population 2 .
A. Inadequacy of the standard Landau-Zener approach for the case of latching modulation In the case of latching modulation, the transition frequency is instantaneously and periodically switched between two constant values. In contrast to the previous studies, the adiabatic evolution and the transitions are now clearly separated. This kind of time-evolution can be achieved by using a modulation f (t) with two very different timescales: one very slow, realizing the simplest adiabatic evolution in a time-independent form for a time 2π/Ω where Ω is the angular frequency of the modulation; and the other one very fast, corresponding ideally to a sudden change in the frequency of the qubit. Since in-between the sudden transitions the system is 'latched' to one of the transition frequencies, we will refer to such modulation as periodic latching modulation. In practice, the latching modulation can be created with a square wave function (50% duty cycle), Conceptually, this is very different from the conventional LZS problem. Let us recall the Landau-Zener (LZ) formula: where the rate of change of the diabatic energy separation evaluated at the crossing is given by v LZ = [df (t)/dt] cross . A direct application of this formula for the case of ideal sudden latching would correspond to an infinite LZ velocity v LZ = ∞. Then the Landau-Zener formula yields p LZ = 1, predicting the absence of interference and therefore constant qubit population 2 . The reason for the inadequacy of the LZ approach for the problem of latching is that this formalism requires the asymptotic match to the adiabatic states far enough from the avoided crossing point. In our case this assumption is not satisfied, since δ can be finite. However, as we will see below, one can indeed recover the LZ result in the asymptotic limit δ → ∞.
Thus we have to develop a novel theoretical model suitable for the case of periodic latching modulation. This is done next by employing the so-called adiabatic-impulse method.
B. Unitary evolution of a two-level system under periodic latching modulation: exact results Let us consider that the latching period starts from right (r) latch position, where the transition frequency is ν + δ. The other latch position is referred to as 'left' ( ) and the corresponding energy gap is ν − δ, see Fig. 1 (a). The Hamiltonian is diagonalized straightforwardly in both latches, leading to the eigenenergies (r) and the corresponding eigenstates |ψ We can switch from the eigenbasis of the 'right latch' to that of the 'left' by making a unitary transformation where − . Also, sinceĤ is symmetric, p s can be taken to be real. Notice that 1 − p s can be seen as a measure of orthogonality of the two bases, while p s represents the sudden-switch transition probability. Naturally, U →r = U −1 r→ = U T r→ . When the system is switched from one latch to the other, we assume that it does not have time to react by adjusting its state. Accordingly, the (instantaneous) unitary timeevolution during the switching is given by U r→ or U →r . The validity of this sudden approximation is studied in detail in the Appendix.
In-between the switches, the system is "parked" in either of the latches or r, and it gathers adiabatic phase in the corresponding eigenbasis: where φ (r, ) ≡ . During one period, the time-evolution of a state |Ψ(0) starting from the right latch can be written as where and Starting from the ground state |Ψ(0) = |ψ (r) − , the probability of finding the system in the excited state after one period is given by The structure of this equation resembles Stückelberg's single-period population in the LZS-model 2 , with p s playing the role of the Landau-Zener probability. However, unlike the case of conventional continuous modulation, the probability p s does not depend on the frequency of modulation. Moreover, as already pointed out, the Landau-Zener result for the linear switching yields the incorrect result P + = 0, while in our case the probability p s is not necessarily 1, allowing distinct evolutionary paths that can interfere. The Landau-Zener result can still be recovered in the limit of large driving amplitude δ |ω 0 − ω|, g, in which case we can neglect the effects due to g, resulting in p s = 1. The inadequacy of the LZ approach for this problem is due to the assumptions needed in the derivation of the LZ transition formula, which is an asymptotic result and, consequently, should hold exactly only in the case of large driving amplitudes. When the modulation is of the latching form, the deviations from the asymptotic result turn out to be important in the finite amplitude range, resulting in a characteristic interference pattern.
After n periods, one has 26 with Thus, the excited state population after n periods is By averaging over n 1 periods, we obtain the timeaveraged excited state population where |γ| 2 + |α| 2 = 1.

C. Resonances
The maximum excited state population, i.e. a resonance, is obtained when Imα = 0: This can be analyzed further in the regimes p s ≈ 1 or p s ≈ 0, resulting in the conditions respectively. The first resonance condition is thus valid for relatively large values of δ when compared to g and |ν|, while the second one is valid when δ |ν|. It is, however, instructive to plot them in the entire range of δ (see Fig. 3). Let us note that in the case of sinusoidal or triangular modulation one obtains a resonance condition similar to Eq. (19) (see Ref. 28) but, in contrast to those, in our case the validity regimes of Eqs. (20)(21) do not depend on the value of the modulation frequency Ω.
In addition to these resonances that originate from the cyclic evolution, a third resonance condition arises from maximizing P + from Eq. (12), which describes the interference after one period. We get Coherent destruction of tunneling 29 occurs at the locations of anti-resonance, i.e. when Here, m + , m − , n, n are integers. Note also that the average steady state population Eq. (18) depends on the starting latch. The steady state occupation can be obtained by averaging over all possible initial phases of the modulation pulse 30 . Nevertheless, we are mainly interested on the locations of the resonances, which remain invariant under the averaging.
The theoretical description presented here is valid everywhere in the parameter space (δ, Ω). However, a number of analytically intuitive results can be obtained when Ω δ, a limit called the rotating-wave approximation (RWA) regime 28 , see Fig. 1 (b). These results are presented in Section IV. In the following section we give details of the experimental realization of the above scheme.

III. EXPERIMENTAL REALIZATION
We have studied the above scheme in the conventional circuit-QED setup 20 , which consists of a capacitivelyshunted Cooper pair box (a transmon) coupled to a coplanar waveguide resonator used for dispersive readout. The periodic latching modulation is created by feeding a square pulse current, generated by an arbitrary waveform generator, through the flux bias coil coupled inductively into the SQUID loop used to tune the Josephson energy of the transmon. A schematic of the circuit and an optical image of the sample is presented in Fig. 2. In the following, we will show that this results in an effective two-level Hamiltonian with periodic latching modulation, thus realizing the Hamiltonian studied in the previous section.
We start with the full Hamiltonian of the transmon, including the coupling with the resonator: where E C = e 2 /2C Σ is the single-electron charging energy, C Σ is the total parallel capacitance (including the shunt), E J1 and E J2 are the Josephson energies of the two Josephson junctions, and n g is the effective offset of the number of Cooper pairs. The resonator frequency is ω r = 1/ √ L r C r andâ denotes the annihilation operator of the resonator mode. Also, V 0 rms = ω r /2C r and β = C g /C Σ 20 . In the following, we will concentrate on the bare qubit part consisting of the three last terms in the above Hamiltonian.
The Hamiltonian (24) results from the circuit quantization of the qubit coupled to the resonator. As usual, ϕ 1 and ϕ 2 denote the gauge-invariant phase differences across the two junctions, and they fulfill the fluxoid quantization condition Here Φ 0 = h/2e is the flux quantum, and Φ is the total magnetic flux through the loop, which is the sum of the external bias flux Φ ext and the screening flux Φ s . Normally the loop inductance of a transmon is negligibly small, therefore Φ ≈ Φ ext , and, to simplify the notations, we take the flux Φ ext ∈ [−Φ 0 /2, Φ 0 /2]. We defineφ ≡ (φ 1 +φ 2 )/2 , E JΣ ≡ E J1 + E J2 and assume that the transmon asymmetry d ≡ (E J2 − E J1 ) /E JΣ 1. The transmon is flux-biased at a constant value Φ dc , on top of which we overlap the time-dependent square pulse flux: Φ ext (t) = Φ dc + Φ sq (t). As a result, the transmon part of Hamiltonian (24) can be written aŝ It is convenient to introduce the standard harmonic oscillator creation and annihilation operatorsb,b † associated with the operatorsφ andn, where In order to minimize the effects of charge fluctuations, the constant flux bias is chosen so that the zero-point phase fluctuations are small, ϕ zpf 1. In this case, the effective offset charge n g can be eliminated by making a gauge transformation, similar to Ref. 20. Since the phase is localized with only small fluctuations around the equilibrium position, the local minima of the cosine potential cos ϕ can be well approximated by a fourth order polynomial.
The Hamiltonian operator of the qubit part is then written aŝ where the plasma frequency ω p is defined as The first two terms in Eq. (29) comprise the conventional transmon Hamiltonian: a harmonic oscillator with a quartic perturbation. The latter two terms are due to the time-dependent flux modulation. We will show that in the case of square pulse modulation, these terms will result in the periodic latching modulation of the qubit. In terms of the unperturbed harmonic oscillator states {|j }, we obtain j|(b+b † ) 2 |j = 2j+1 and j|(b+b † ) 4 |j = 6j 2 +6j +3; also the even powers ofφ do not couple states with different parity. By truncating Hamiltonian (29) to the Hilbert space spanned by the two lowest energy levels {|0 , |1 }, we obtain where the transition energy and the longitudinal drive f sq (t) = E JΣ 2 sin(πΦ dc /Φ 0 )(ϕ 4 zpf −2ϕ 2 zpf ) sin(πΦ sq (t)/Φ 0 ). (33) By comparing with Eq. (2), we can identify the latching modulation amplitude δ = E JΣ sin(πΦ dc /Φ 0 )(ϕ 4 zpf − 2ϕ 2 zpf ) sin(πΦ sq /Φ 0 ), where Φ sq is the square wave amplitude of the magnetic flux in the transmon SQUID loop.
Besides the flux modulation, the qubit is also driven via the resonator. By neglecting the quantum fluctuations of the resonator, the second term in Hamiltonian (24) can be written aŝ where n r is the number of coherent quanta in the resonator, ω is the driving frequency, and in the latter equality we have made the two-state truncation and defined g ≡ (2βeV 0 rms /ϕ zpf ) √ n r . The vacuum Jaynes-Cummings coupling to the first transition is defined by g 0 ≡ βeV 0 rms /ϕ zpf , thus g = 2g 0 √ n r .
Next, we transform into a frame rotating at the driving frequency ω around the z-axis, implemented by the unitary transformation exp[−iωσ z t/2]. With an additional rotationσ y →σ x , we obtain finally the effective Hamiltonian: which defines a σ x -coupled qubit with frequency ν = ω 0 − ω modulated by f sq (t). In other words, in our experiment the generic Hamiltonian (1) is realized as an effective Hamiltonian in the subspace of dressed states formed by the qubit and the transverse driving field. The readout of the transmon is based on the ac Stark shift of the resonance ω r from its bare frequency ω r /2π = 3.795 GHz, resulting in a change in the microwave reflection coefficient S 11 . The dependence of the energy level separation on the applied external magnetic flux given by Eqs. (30) and (32) can be used to extract the transmon parameters. We have diagonalized the full transmon Hamiltonian, and by fitting with the measured spectrum (white dashed line in Fig. 2) we can extract E C /h = 0.35 GHz, E JΣ /h = 8.4 GHz, and d ≈ 0.1. The relaxation rate Γ 1 /2π = 1.2 MHz and the dephasing rate Γ 2 /2π = 3.1 MHz were obtained by independent characterization measurements 31 . The value g 0 /2π ≈ 80 MHz for the Jaynes-Cummings coupling between the resonator and the transmon was extracted from vacuum Rabi mode splitting data. To allow a good fidelity in the transmission of the square pulse at relatively high frequencies, the qubit was minimally filtered, with the downside of an increased noise level. Next, we show the results of the periodic latching modulation experiment and give an alternative interpretation of the data in terms of the rotating wave approximation (RWA).

A. Comparison between the simulations and the experimental results
Here we compare the experimental results with numerical simulations of the dispersive shift. In Fig. 3 we show the experimental results and the numerical simulations for the latching modulation, together with the resonance conditions (20), (21), and (22). In the simulation we have set the temperature of the environment to T = 50 mK, which is the base temperature of the refrigerator. To take into account the thermal excitations properly, the Hamiltonians (31) and (34) are extended for the five lowest transmon eigenstates 20,32 . We solve numerically the steady state population of the driven and modulated five level transmon in a thermal bath by using the quantum trajectory method 33 . We assume that the driving field couples only the ground state and the first excited state. This approximation is reasonable since the detuning of the field is smaller than the anharmonicity E C , that is |ω 0 − ω| < E C . In the dispersive limit 20,32 , the transmon population shifts the eigenfrequency of the resonator by where P i is the steady state population of the i:th level and  We note that both the experimental and the numerical resonance locations are in excellent agreement with the analytic latching modulation model. The resonances predicted by Eq. (20) and Eq. (21) coincide remarkably well, when they are within their range of validity, with those from the numerical and the experimental results. We also note that due to reduced filtering, most likely the noise level felt by the qubit is higher than in the ideal situation. Effective qubit temperatures larger than 100 mK have been determined previously in transmons that do not thermalize properly 34 . Indeed in our simulations we find that by increasing the temperature to higher values results in reduced contrast of the fine structures of Fig. 3(a), in accordance with the experiment.
Also, it is possible to measure the qubit population as a function of detuning by varying the modulation amplitude at a fixed modulation frequency, which is the standard representation of the LZ interference. However, it is known that in this representation the differences between sinusoidal and other types of modulation are not so well visible 12 , and indeed our data confirms this result. In Fig. 4 we plot a few population oscillations as a function of modulation amplitude δ and detuning ν = ω 0 −ω at the fixed modulation amplitude Ω/2π = 50 MHz. Although some interference pattern is visible, there is no distinctive difference with respect to the patterns obtained for sinusoidal or triangular modulation in standard LZS experiments 2 .  Fig. 3.

B. Comparison between periodic latching modulation and sinusoidal modulation
Now we compare the spectrum of the system under the periodic latching modulation with that of a sinusoidal modulation with exactly the same parameters (the same qubit frequency ω 0 and the same modulation amplitude as in Fig. 3), shown in Fig. 5.
One notices already in Figs. 3 and 5 that the spectral structure at low and intermediate modulation frequencies is rather different. To illustrate the differences, in Fig. 6 (a) we present a comparison between the sinusoidal and the latching modulation along the second sideband, where for clarity we show the spectra in the low-frequency range, up to 80 MHz. Here, in order to eliminate the asymmetry seen in the cavity response between positive and negative ν = ω 0 − ω, we calculate the average of the sidebands m = −2 and m = 2. The rather poor signal to noise ratio does not allow us to clearly identify all the population oscillations, but some differences can be seen clearly.
Best seen in Fig. 6 (a), the latching resonance at around Ω/2π 45 MHz is clearly shifted towards higher value of Ω when compared to that of the sinusoidal modulation. Also, at the low-Ω end the dispersive shift due to the sinusoidal modulation remains around ∆ω r /2π ≈ 0.8 MHz when Ω is decreased, whereas in the case of the latching modulation there is a considerable drop (ideally to zero). This can be seen also from the spectra in Figs. 3 and 5, and is due to the fact that in the periodic latching modulation at low frequencies the qubit spends almost no time at ω 0 − ω = 0. We have confirmed this behaviour with several other values of δ. As the frequency increases, sidebands start to form and the signal corresponding to periodic latching modulation increases to values close to that of the sinusoidal modulation.

C. Slow-modulation and fast-modulation (RWA) regimes
The existence of two regimes, a slow-modulation regime for Ω δ/2 and a fast-modulation regime for Ω δ/2 is quite apparent in Figs. 3 and 5. In the slow-modulation regime a fine structure of resonances appear and the differences between the latching modulation and other types of modulation become visible, while in the fastmodulation regime the sidebands are the prominent feature. In order to get a better qualitative understanding of the fast-modulation regime, we develop here an alternative analytic description in terms of the two-level Hamiltonian (35) and the rotating wave approximation. This description is valid at relatively large modulation frequencies Ω g, shown schematically in Fig. 1 (b).
We transform the Hamiltonian Eq. (35) into a frame co-rotating with the longitudinal modulation f (t) by employing the unitary transformation which, in the Bloch-sphere picture, corresponds to a frame rotating around the z-axis with the instantaneous angular velocity f (t). In this frame, the new effective Hamiltonian is obtained byĤ =Û †ĤÛ + i (∂ tÛ † )Û , where A(t) = exp i t 0 f (τ )dτ . We use the Jacobi-Anger relation 35 to find the harmonic-mode expansion for the effective transverse drive A(t): The sideband amplitude for periodic latching f (t) = f sq (t) can be written in the form where all k j 's take integer values from −∞ to ∞ and the summation goes over all possible combinations {k j } that result in m = When ω ≈ ω 0 + mΩ, and if the other driving fields are not too strong g|∆ sq k | < Ω, k = m, we can make the rotating wave approximation (RWA) by neglecting the non-resonant driving fields. When Ω is small, the RWA results below can be improved by adding Bloch-Siegert and higher order corrections (so-called generalized Bloch-Siegert shift 36 ).
We can find the steady state occupation probability P e by solving the Lindblad form master equation analytically around every resolvable resonance 15,37 . The master equation including the pure depahsing and the energy relaxation processes, with rates Γ ϕ and Γ 1 , respectively, is written for two lowest transmon levels using the Hamiltonian Eq. (35) Note that when the decoherence is introduced, the widths of the sidebands are broadened due to both the decoherence rate Γ 2 = 2Γ 1 + Γ ϕ and the power broadening caused by the strong transverse driving, g∆ sq m 37 , yielding a linewidth Γ 2 2 + (g∆ sq m ) 2 Γ 2 /Γ 1 . By adding the contributions from all resolvable resonances, we get for the steady state occupation probability of the qubit excited state.
In Fig. 6 b) we present the results of the RWA method for the second sideband m = 2. In this figure we use Eq. (43) and we truncate the series Eq. (40) to j max = 5. The results of the RWA Eq. (43) are in reasonably good agreement with the numerical ones down to Ω/2π ≈ 20 MHz (continuous green line). Below this value we see deviations as the rotating wave approximation becomes inadequate for reproducing the numerical data. Nevertheless, the RWA predicts relatively well the position of the resonances (dashed green line). Note that in the experiments we can reach a wide range of values for δ/Ω, for example in the spectrum Fig. 3, the ratio δ/Ω = ∞ . . . 1, while in the standard representation Fig. 4 we have δ/Ω = 0 . . . 5.
The RWA analysis explains also why the first maximum of the periodic latching modulation is shifted towards higher values compared to the sinusoidal, as noted in the previous subsection, see Fig. 6 (a). In the extreme asymptotic limit, Ω → ∞, one can neglect all but m = k 0 terms in the sum, resulting in ∆ sq m ≈ J m (4δ/πΩ). Because 4 > π, one expects that the first resonance (maximum of ∆ sq m ) will be shifted towards higher values of Ω when compared to the sinusoidal case ∆ sin m = J m (δ/Ω). This can be seen also in our experimental data for m = 2, displayed in Fig. 6 (a).

V. CONCLUSIONS
We have shown that the periodic latching modulation of the transition frequency of a qubit is conceptually different from the continuous drive forms used in the conventional studies of LZS-interference. We have adapted the adiabatic-impulse method for the case of abrupt and periodic switching, and the results are shown to be in good agreement with more elaborate numerical calculations. We have studied the periodic latching modulation experimentally by employing a transmon with flux bias modulated with a square pulse pattern. We measured a spectrum where two regimes (slow-modulation and fastmodulation) can be clearly distinguished. The spectrum has a rich structure of sidebands, due to resonances and anti-resonances (coherent destruction of tunneling). The experimental data were in good agreement with our theoretical models, and we were able to extract the information about the pulse shape from the region of low modulation frequency. Our results open the way for simulating various forms of dephasing noise and for realizing experiments where the switching of the qubit frequency has a specific, non-sinusoidal time dependence.